1. Vibrational (Infrared) Spectroscopy
vibrational modes
← C≣≣O →
equilibrium bond distance re can be changed by
applying energy
potential well for modified (Morse) potential
classical vibrator well for a diatomic molecule
1. quantized – only certain energy levels may exist
E = hω(u + 1/2) 
u: vibrational quantum number
w: vibrational frequency k
ω = ―― ――
2p m
m1 × m2
m = ―――――
m1 + m2 
  2. too close – repulsion between nuclei and electrons
too far apart – dissociation

2. ex. HCl u(HCl) = 2990 cm-1
DCl u(DCl) = 2145 cm-1
ex u(NO) bond order
NO cm-1 3
NO 1880 cm-1 2.5
NO cm-1 2
NO cm
number of vibrational modes
a molecule consists of N atoms, there are 3N
degrees of freedom
translation rotation vibration
nonlinear N – 6
linear N – 5
type of vibrational modes
stretching mode u bending mode d
IR active absorption Raman active absorption

3. frequencies for some commonly encountered groups, fragments,
and linkages in inorganic and organic molecules

4. ex. W(CO) Mn(CO)5Br
compound u(CO) (cm-1)
[Ti(CO)6]
[V(CO)6]
Cr(CO)
[Mn(CO)6]
stretching modes of CO and IR frequencies
(a) terminal (b) doubly bridging (c) triply bridging
   ex.

5. some ligands capable of forming linkage isomers
  IR spectrum for nujol
  salt plates
NaCl cm-1
KBr cm-1
CsI cm-1
721
1377
1462
2925
2855

6. symmetry of normal vibrations
ex. CO vibrational modes
C3(u3a) = -1/2u3a + 1/2 u3b
C3(u3b) = -3/2u3a - 1/2 u3b
determine the symmetry type of normal modes
  E
c = 12

7. C3
  c = 0
  C2
c = -2
c (sh) = 4
c (S3) = ??
c (sv) = 2

8. G = A1’ + A2’ + 3E’ + 2A2” + E”
3 translatory modes: E’, A2”
3 rotational modes: A2’, E”
genuine vibrational modes: Gg = A1’ +2E’ + A2”
IR active: E’, A2” (3 bands)
Raman active: A1’, E’ (3 bands)
particular internal coordinates to normal modes
C—O bonds
E C C sh S sv G
GCO = A1’ + E’ in-plane stretching
GOCO = A1’ + E’ in-plane bending
A2” out-of-plane bending
ex. determine the number of IR active CO stretching
bands for the following metal carbonyl
compounds : M(CO)6 M(CO)5L cis-M(CO)4L2
trans-M(CO)4L2 fac-M(CO)3L3 mer-M(CO)3L3
M(CO)5 M(CO)4L M(CO)3L2 M(CO)4

9. (i) trans-M(CO)4L2 D4h E C4 C2 C2’ C2” i S4 sh sv sd
OC CO
OC CO ==> A1g + B1g + Eu
L IR-active: Eu
(ii) cis-M(CO)4L2
CO C2v E C s s’
OC L
OC L ==> 2A1 + B1 + B2
CO IR-active: 2A1, B1, B2
(iii) mer-M(CO)3L3
CO C2v E C s s’
L L
OC L ==> 2A1 + B1
OC IR-active: 2A1, B1
(iv) M(CO)5
D3h E C C2 sh S3 sv
==> 2A1’ + A2” + E’
IR-active: A2”, E’

10. (v) M(CO)4L L D3v E C3 sv 4 1 2 ==> 2A1 + E IR-active: 2A1, E
==> 2A1 + E
IR-active: 2A1, E
D2v E C sv sv‘ L
==> 2A1 + B1 + B2
IR-active: 2A1, B1, B2
(vi) M(CO)3L2
L D3h E C3 C2 sh S3 sv
==> A1‘ + E’
L IR-active: E’
L Cs E sh
L ==> 2A‘ + A”
IR-active: 2A’, A”
(vii) M(CO)4
Td E C3 C2 S3 sd
==> A1 + T2
IR-active: T2

11. number of CO stretching bands in IR spetcrum for metal carbonyl compounds

12. number of IR bands of some common geometric arrangements

13. calculation of force constants
for diatomic molecule AB harmonic oscillator
f • m-1 – l = 0
for polyatomic molecule
Wilson’s method “The F and G matrix method”
|FG – El| = 0
F: matrix of force constant (potential energy)
G: matrix of masses and spatial relationship of
atoms (kinetic energy)
E: unit matrix 
e.g. H2O Gg = 2A1 + B1
2 O-H distance Dd1, Dd A1 + B1
∠HOH Dθ A1
using projection operator to obtain complete set
of symmetry coordinates for vibrations
A1 : S1 = Dθ
S2 = 1/√2(Dd1 + Dd2)
B1 : S3 = 1/√2(Dd1 - Dd2)
F matrix 2V = Sfik si sk
si, sk: change in internal coordinates
for Dd1 Dd2 Dθ
Dd1 fd fdd fdθ
Dd2 fdd fd fdθ
Dθ fdθ fdθ fθ

14. 2V = fd(Dd1)2 + fd(Dd2)2 + fθ(Dθ)2 + 2 fdd(Dd1 Dd2)
+ 2 fdθ(Dd1 Dθ) + 2 fdθ(Dd2 Dθ)
= [Dd1 Dd2 Dθ] fd fdd fdθ Dd1
fdd fd fdθ Dd2
fdθ fdθ fθ Dθ
= s’f s
relationship between the internal coordinates and
the symmetry coordinates
S = U s
U matrix
Dq Dd1
Dd1 + Dd2 = 1/√ /√ Dd2
Dd1 - Dd /√ /√ Dq
S = U s s = U’ S s’ = (U’ S)’ = S’U
s’fs = S’FS
(S’U)f(U’S) = S’FS
S’(UfU’)S = S’FS ==> F = UfU’
fd fdd fdθ /√2 1/√2
F = 1/√2 1/√ fdd fd fdθ /√2 -1/√2
1/√ /√ fdθ fdθ fθ
fθ √2 fdθ
= √2 fdθ fd + fdd
fd - fdd

15. G = 1/√2 1/√2 0 gdd gd gdθ 0 1/√2 -1/√2 G matrix G = UgU’
gd gdd gdθ /√2 1/√2
G = 1/√2 1/√ gdd gd gdθ /√2 -1/√2
1/√ /√ gdθ gdθ gθ
g √2 g13 0
= √2 g g11 + g12 0
g11 - g12
g11 = mH + mO
g12 = mO cosθ
g13 = -(mO/r) sinθ
g33 = 2(mH + mO - mO cosθ)/r2
m : reciprocal of the mass
2(mH + mO - mO cosθ)/r2 -(√2mO/r) sinθ
G = (√2mO/r) sinθ mH + mO (1+ cosθ)
mH + mO (1 - cosθ)
for H2O θ= 104.3o31’ r = Å
G =
fθ √2 fdθ l
A1: – = 0
√2 fdθ fd + fdd l
B1: (fd - fdd) = l

16. elements of the g matrix
mi: reciprocal mass of the ith atom
rij: reciprocal of the distance between ith and jth

17. Raman spectroscopy
light of energy less than that required to
promote a molecule into an excited electronic
state is absorbed by a molecule,
a virtual excited state is created
virtual state is very short lifetime, the majority
of the light is re-emitted over 360oC, this is
called Rayleigh scattering
C. V. Raman found that the energy of a small
proportion of re-emitted light differs from the
incident radiation by energy gaps that
correspond to some of the vibrational modes
Stokes lines
anti-Stokes line

18. schematic representation of Raman spectrometer
selection rules for vibrational transitions
• a fundamental will be infrared active if the
normal mode which is excited belongs to the
same representation as any one or several of
the Cartesian coordinates
• a fundamental will be Raman active if the
normal mode involved belongs to the same
representation as one or more of the
components of the polarizability tensor of the
molecule
the exclusion rule – in centrosymmetric molecules,
no Raman-active vibration is also IR-active and
no IR-active vibration is also Raman-active
only fundamentals of modes belonging to g
representations can be Raman active and
only fundamentals of modes belonging to u
representations can be IR active

19. ex. Na2MoO4 dissolved in HCl exhibits Raman
peaks at 964, 925, 392, 311, 246, 219 cm-1
925, 311 cm-1 being polarized
what can be deduced from the spectrum?
no n(Mo—H) and n(O—H) bands
only M—Cl and M=O likely exist
964, 925 cm Mo=O stretching bands
392 cm Mo=O bending mode
311, 246, 219 cm-1
Mo—Cl stretching modes
possible product:

20. normal vibrational modes for common structures